Final answer:
To balance the electrostatic repulsion with the gravitational force, you would need to remove approximately 1.161 × 10^27 electrons from each of the two similar spheres.
Step-by-step explanation:
To balance the electrostatic repulsion with the gravitational force, we need to remove a certain number of electrons from each sphere. Let's calculate step by step:
1. Determine the charge on each sphere:
- Sphere 1: It has a charge of -9.6 × 10^-18 C.
- Sphere 2: It has 30 excess electrons. The charge of each electron is -1.6 × 10^-19 C, so the total charge on Sphere 2 is (-1.6 × 10^-19 C) × (30 electrons) = -4.8 × 10^-18 C.
2. The electrostatic repulsive force between two charges is given by Coulomb's law:
F = (k * |q1 * q2|) / r^2
where F is the force, k is the electrostatic constant (9 × 10^9 N m^2/C^2), |q1| and |q2| are the magnitudes of the charges, and r is the distance between them.
3. The gravitational force between two objects is given by Newton's law of gravitation:
F = (G * |m1 * m2|) / r^2
where F is the force, G is the gravitational constant (6.67 × 10^-11 N m^2/kg^2), |m1| and |m2| are the magnitudes of the masses, and r is the distance between them.
4. Set the forces equal to each other:
(k * |q1 * q2|) / r^2 = (G * |m1 * m2|) / r^2
5. Cancel out common terms:
k * |q1 * q2| = G * |m1 * m2|
6. Solve for q1:
q1 = (G * |m1 * m2|) / (k * |q2|)
7. Calculate q1:
q1 = (6.67 × 10^-11 N m^2/kg^2 * (10 g * 10^-3 kg / (1 g)) * (10 g * 10^-3 kg / (1 g))) / (9 × 10^9 N m^2/C^2 * 4.8 × 10^-18 C)
q1 ≈ 1.858 × 10^8 C
8. Calculate the number of electrons equivalent to the charge on each sphere:
Number of electrons = q / charge of an electron
- For Sphere 1: Number of electrons = 1.858 × 10^8 C / (1.6 × 10^-19 C) ≈ 1.161 × 10^27 electrons
- For Sphere 2: Number of electrons = -4.8 × 10^-18 C / (1.6 × 10^-19 C) ≈ -30 electrons
Therefore, to balance the electrostatic repulsion with the gravitational force, approximately 1.161 × 10^27 electrons need to be removed from each of the two similar spheres.